Dedicated to the application of mathematical modeling and fieldwork to the study of the history of cartography...and perhaps a few philosophical reflections on the above...

Author Climbing in the Queyras, Summer 2013

Friday, May 20, 2011

The Morphology of the Early New England Landscape:Searching for the Lost Maps of Henry David Thoreau

New York Times/Mattson LectureOctober 16thOsher Map LibraryUniversity of Southern Maine

An article based on this lecture and intervening research will be pubished in the March 2011 issue of the Concord Saunterer: A Journal of Thoreau Studies.

The human organism has rarely been subjected to a severer test than the study of scientific problems, nor is there a truer hero than an investigator who never loses heart in a life-long grapple with the powers of the universe. It requires courage of the highest order to stand for years face to face with one of the enigmas of nature; to interrogate patiently, and hear no answers......................................................................................................................---Clarence King

Synopsis of the text of my lecture at the re-opening of the Osher Map Library and the "New Directions in the Study of Early American Cartographies" Conference...

In April of 1858, Ralph Waldo Emerson, in a letter to his friend H.S. Randall, wrote that, “Thoreau’s study seems at present to be equally shared between natural and civil history,” and that “he reads both with a keen and original eye.”

The civil history that Emerson refers to here is the history of the early exploration and discovery of the North American continent, especially the northeastern coast of New England and Canada. During the last 12 years of his life, from about 1850 thru 1862, Henry David Thoreau dedicated himself to historical and scientific studies that have either been ignored by or have puzzled generations of his commentators. What was the author of Walden and other works of transcendental literature doing out “in all weathers” as Emerson would say, counting tree rings, measuring the differences in the magnetic variation of compass needles, mapping the depths of streams or listing the blooming of plant species. Why was he borrowing the earliest exploration narratives of the New World from the Harvard Library, taking detailed notes on the names of places and the plants and animals mentioned, and making scaled copies of some of the earliest maps of North America?

During these last, but extremely productive, years of Thoreau’s life his interests turned sharply toward these types of more empirical and less transcendental studies—Thoreau being less influenced in his work at this time by Emerson than by the geographically oriented science of Humboldt and Darwin. Thoreau believed that the secrets of nature, and of humanities place within it, were ultimately revealed by identifying what was significant in the everyday world and that this in turn depended on meticulous attention to and an accounting of, the commonplace.

In this spirit Thoreau’s writings such as the Maine Woods, Cape Cod, Walden, A Yankee in Canada, his natural history essays, and of course his journals have occasionally been probed by humanist geographers and linked with the beginnings of modern environmental thought…this linkage stems mostly from Thoreau’s intense concern with the concept of place and his ability to see deep connections between historical process and environmental change.

Tonight I am going to talk to you about this linkage in a slightly different way than those geographers and historians who have probed Thoreau’s writings…for tonight rather than concentrate on his published works…I am going to speak with you about his cartography and how Thoreau’s cartographic explorations provided a link in his mind between natural and civil history… a link that led him to a very modern sense of man’s place in nature. To do this I am going to discuss, in more detail than perhaps has been done before, several important aspects of Thoreau’s mostly unknown and certainly understudied cartographic worksThe first aspect, and perhaps the most important for his technical understanding of cartography and the process of mapmaking, was his work as a land surveyor. This work gave Thoreau the ability to look at maps critically and to understand not only their mathematical limits but also their broader cultural meaning. It also allowed him to wander the fields and woodlots of Concord and to observe nature closely in all seasons…in a way that his fellow transcendentalists certainly never would have.

The second aspect, and for the most part the driving reason for his engagement with early American cartography, was his interest in the history of the indigenous peoples of the north east. Thoreau would take nearly 4000 pages of notes on this subject…none of which has yet to be published and all of which are mostly unknown even to Thoreau specialists. In the course of this note taking Thoreau would copy and comment on many of the early maps of the northeastern United States and Canada by such seminal figures as Champlain, Ortelius, Smith, Verrazano and Wytfliet.

Finally, I am going to present to you two new maps that have recently been attributed by me to Thoreau and discuss my search for some of the other missing maps that he copied and annotated.

Some of what I am about to say might sound to some of you slightly academic, but really my talk tonight it is about something quite humble and speaks to the very heart and reason of why we are all here this evening. What am speaking about in the end is a story about a man in a Library, much like the one we celebrate this weekend, with a notebook, some old maps and his imagination on fire…

I am going to begin with one of those notebooks, namely Thoreau’s field notebook in which he kept the rough notes from his surveying activity. Even though we are talking tonight about Thoreau’s maps, it is with a pencil and a notebook that he truly found his way in the world…and we will have reason this evening to look at many of Thoreau’s notebooks.

....The field notebook appears on the surface uninteresting as it is filled for the most part with measurements and locations…places around Concord, MA that Thoreau surveyed. But it provides a detailed record not only of how Thoreau worked but also how he approached the more technical aspects of cartography and we shall return to this notebook many times this evening. Thoreau surveyed many places around Concord and the list of his clients reads like a library of early American authors. Places like Bronson Alcott’s farm shown here.

....

Thoreau set himself up as a surveyor in late 1849 and by the end of 1851 was recognized as one of best and most accurate operating in the region. The question of exactly how Thoreau learned to survey is for the most part an open one but it does appear that he was mostly self taught. In early April of 1850 Thoreau would borrow from the Harvard Library a copy of Davies’ Elements of Surveying and navigation , [we will see a great deal of Library use by Thoreau this evening.] A copy of the 1847 edition of Davies was also owned by Thoreau. Davies was one of the most widely used books on surveying during the middle of the 19th century and with Galbraith’s mathematical tables, also borrowed from Harvard, represents a solid introduction to the principles of land measurement. Thoreau annotated his copy of Davies with notes from Galbraith on particular mathematical problems…this is Galbraith’s instructions on how to calculate the sine of an angle.

Thoreau took much more from Galbraith than mere mathematical instruction. In a journal entry dated June 9th of 1850 Thoreau lists nine books recommended by Galbraith’s text regarding the esoteric subject of the magnetic variation of compass needles. I am going to spend some time with this subject because I believe it shows how Thoreau thought through and imagined the complexities of mapmaking and his engagement with the subject can be used as a model for us to think through his transition from an Emersonian transcendentalist…to the more empirical view of the world that informed his readings and use of early American cartography

Thoreau’s interest in magnetic variation is first indicated on his advertising broadside where he explains the variation of the compass is noted so that the survey can be repeated.Throughout the early 1850’s one finds references throughout his journals and field notebooks made to the books and articles Thoreau read on the subject and to the observations of compass needle variations that he made in and around Concord.

Magnetic declination for those of you who do not know is the variation that we see between magnetic north, or the north that a compass needle points to, and true north, the direction of the pole. The variation in the compass needle is caused by the earth’s magnetic field and was the subject of a great deal of scientific research in the mid-19th century. The exact direction that a compass needle points to is not constant even for a specific location, and although few people would notice these small changes, Thoreau, noted them quite explicitly…for example…in November of 1850, he made an entry in his journal that marks the beginning of what would be almost an obsession with the subject…

Thoreau writes,

“When I am considering the way which I walk, my needle is slow to settle, my compass varies by a few degrees and does not always point due southwest; and there is good authority for these variations in the heavens…”

Thoreau’s interest is more than just passing and he delves into the science of magnetic declination in a way that would become representative of his work not only in natural history but also in the way he approached cartography as well. In his field notebook he explains how he established the True meridian so he could continually check his surveys against the variation of his compass needle

True meridian slide

• Found the direction of the pole star at its western elongation (1,58-1/2) at 9h 26m PM (Feb 7th 1851).• N coincides with a line drawn from the SE course of the stone post on the E side of our western small front gate, to the S side of the first door on the W side of the depot.

Thoreau has measured a reference line for the direction of true north… from the west gate of his home in Concord to the depot across the street.By establishing a sight line for the True Meridian from his family’s house to the depot, Thoreau could easily check the declination of his compass before or after surveying. Thoreau would begin to include this information on his surveys even though it makes little difference to the purpose of the survey itself. For example we can see on his survey of Hosmer’s farm that Thoreau has added compass headings to each of the boundaries.

Thoreau would continue for some time to make detailed observations of magnetic declination and write them into in his field notebooks…He would also go as far as to contact and correspond with William Cranch Bond, the director of the Harvard Observatory. Bond was conducting experiments in magnetic variation in Cambridge, and took thousands of measurements on magnetic declination in order to try to predict the changes that he and Thoreau saw in compass needles, a phenomenon that we now know to be chaotic. One can see just how chaotic by looking at one of the many graphs of Bond’s published measurements that Thoreau certainly read.

Thoreau soon found that Bond’s results were not applicable to Concord, even though it was just a few miles away.

One of the most amazing and important things about all this material is how it shows a change in Thoreau’s thought process and his turn away from the transcendentalist mode of thought that drove his early works. Among transcendentalists' core beliefs, at least as it was realized by Emerson, was an ideal spiritualstate that 'transcends' the physical and empirical and is only realized through the individual's intuition. In other words a real mind over matter philosophy. Emerson found little of higher worth in the empirical and downgraded most of science as “mere facts”. Thoreau would begin by the early 1850’s to leave these idealist tendencies behind and turn toward more realist studies of nature and history…to the point that by late 1852 he could write in his journal seemingly against Emerson, that “Mere facts & names & dates communicate to us much more than we suspect…”

This empirical turn toward a more scientific world view… would influence all of Thoreau’s thought from around 1852 onwards. Although we can never really call Thoreau a thinker who fully embraced the pure empiricism of late 19th century science, there is a change in his thought that effects all of his reading and observations even his interpretation of the early exploration narratives and the history of cartography… it is to his cartographic explorations that we will now turn…

In order to talk about Thoreau’s interactions with and interest in early American exploration and cartography we must look to what has become known as the “Canadian Notebook”. The notebook was begun shortly after Thoreau’s return from a trip to Canada and those few scholars who have actually read it….it remains unpublished… differ in their opinions of why it was composed. The notebook is divided into three parts… the second part is certainly the most intense interaction with cartography and cartographic literature that one will find in the entire Thoreau corpus.

This second part of the notebook contains notes from Thoreau’s reading of early exploration narratives and maps by such figures as Champlain, Lescarbot, John Smith, Ortelius and Wytfliet. Thoreau takes note of specific subjects like the changing of place names, the plants and animals that the explorers encountered, the size and flow of rivers, temperatures, snowfall, and the changing shape of the lakes and rivers shown on their maps. In other words facts, empirical data that describes the landscape and the conditions of place. This type of description is paralleled in his journal entries at the time where he is noting things like stream depths, tree ring counts, snowfall, and the blooming times of plants that he observed during his surveying of woodlots and farms around the Concord countryside.

Even as Thoreau was taking detailed notes on the maps and information found in these early exploration narratives he expressed his frustration with the study of human versus natural history. In his journal, on October 19, 1860, he writes,

“It is easier far to recover the history of the trees which stood here a century or more ago than it is to recover the history of the men who walked beneath them, How much do we know---how little more can we know—of these centuries of Concord life?”

It was to answer this question that Thoreau turned to early cartography and the texts that accompanied them. In the back of the Canadian notebook written in Thoreau’s hand, but in pencil, and in the wrong direction if one is reading from the front, Thoreau composes the following list of maps that he has copied…

On the back fly leaf written upside down in Thoreau’s hand:“I have copied maps made ac. to…”1.Verarzani’s plot in Hacklyts divers voyage 15822. Map made in forme sent from Seville in 1527 by Thorne3. Map of Nova Francia etc. in Ramurio 3rd volume (1556) ac to discourse of a great sea captiane4. Of America in Ortelius (1570 &e) who used Cabot and others5. Of Norumbega and Virginia 1597 Wytfliet Lovanni6. Nouvelle France Champlain 1612, 1632

The list, although there are 6 numbers, actually describes seven maps there being two by Champlain listed on the same line. I first encountered this list many years ago and became interested in determining if copies of the maps that Thoreau listed still survive, if he in fact drew them, and what they might have looked like.

Two of the maps on the list were quite easy to locate and are part of the Collection of Thoreau surveys and papers in the Concord Free Public Library. The one shown here was copied by Thoreau from Abraham Ortelius’ world atlas, the Theatrum Orbis Terrarum, published in 1570. Ortelius was the royal geographer to King Phillip II of Spain and a prolific atlas maker. Thoreau borrowed several different editions of the atlas from the Harvard Library over a period of years and I have not been able to determine when he made the sketch. Thoreau’s notes on Ortelius in the Canadian notebook are principally devoted to a commentary on this map…”a new description of America”…and they are very much concerned with place names…as is typical of these notebook entries that deal with maps Thoreau brings other sources in and makes comparisons…in the case of Ortelius Thoreau will ask about the antiquity of some of the place names the cartographer uses.

The second map found in the Concord Library is that of Cornelius Wytfleit. Wytfliet was a Flemish geographer who published an atlas called Descriptionis Ptolemaicae Augmentum in 1597. This sketch, which is cruder than the Ortelius, was made in 1855 again from a book borrowed many times from Harvard Library.

The other five maps have been more difficult to find and to identify. Three of them are very small and are found folded in the misc papers of Thoreau that are in the Morgan Library…they are unidentified and therefore do not appear in their catalog lists or finding aids. (an example shown below)

Of all of the notes in the Canadian notebook about cartography, by far the most extensive are associated with the narrative of Champlain’s voyages…time and time again Thoreau will borrow the 1613 and 1632 editions of this book from the Harvard Library…Champlain made several voyages to the New World and explored the St. Lawerence River, along with most of the New England coast, at least as far south as Cape Cod. The narrative of his voyages is filled with maps and his reflections on the explorations.

Thoreau’s notes on Champlain’s texts are full of details on plants, animals, place names and especially the differences in the various editions of the maps Champlain made.…Thoreau writes praising Champlain’s accuracy as a geographer and he quotes in detail Champlain’s own commentary on the methods of his mapmaking.

But what about the two Champlain maps on Thoreau’s list of the maps he copied???…they appear in no inventory of any library, they are talked about in no scholarly articles and appeared to me to be lost.

Then one day, when I first came to the Library of Congress, almost ten years ago I happened to be doing some research on Thoreau and the depth of Walden Pond when Ron Grim…now of the Boston Public Library, mentioned to me that there were several maps in the Geography and Map Division thought to be by Thoreau but with no real attributions. Curious about this but not expecting much…for how could maps by Thoreau be sitting in the Library of Congress without firm attributions…I opened the folder and saw a manuscript…

Immediately I knew that I was looking at the missing Champlain maps. The manuscript that you see here is one of four pages that describes the two Champlain maps that Thoreau listed on the flyleaf of the Canadian notebook and copied, one from 1632,

This map was easy to identify as Thoreau’s because it is in his hand and fits the general form of his other map sketches. The second map however from 1612 was more difficult as it is a tracing and contains almost nothing that would directly link it with Thoreau. Thoreau does say in the manuscripts that accompany the maps that he annotated the 1612 map in red with later places names. This feature can be seen on the 1612 copy below.

While the finding of the two Champlain maps completes the list of the known cartographic copies that Thoreau made… it has opened up the question of how to understand Thoreau’s geographic explorations in relation to some of his larger projects and published works. For Thoreau’s relationship to cartography is a complicated one and has suffered from a lack of scholarly attention. In general Thoreau seems to have remained skeptical of maps even as he made constant use of them…in his journals he wrote,

How little there is on an ordinary map! ...between those lines indicating roads is a plain blank space in the form of a square or triangle or polygon or segment of a circle, and there is naught to distinguish this from another area…for on it are no moral lessons…

And in the Maine Woods he tells us that maps are “labyrinths of error.”

Thoreau’s true attitude toward cartography is not difficult to assess, if one takes the time to read through his extensive notes on the subject closely. The notes express an immediacy of experience that occurs when one is reading and observing directly…Thoreau did not think of historic maps from the past as obsolete, but rather as graphic and ideological documents that could help him understand what had been in a particular place before…In many ways we can consider Thoreau the first "modern" historian of cartography.

To conclude, I want to return to one of Thoreau’s surveys.

....

It is a simple drawing of a woodlot but I think it sums up Thoreau’s relationship to cartography and its influence on his work. The surveying of woodlots was very much part of Thoreau’s daily routine and it took him into areas of Concord seldom seen by his fellow citizens…surveying a woodlot generally meant the lumber it contained was up for sale and it was going to be cut down. Thoreau would return to the lots after they had been cut and in his journals noted in detail the succession of plants and trees that would follow…these notes would result in Thoreau’s most important work of natural history, “The Dispersion of Seeds”, which he composed shortly after reading Darwin in 1860…

Unfortunately, Thoreau did not live long enough to complete the great work on geography and the indigenous peoples of North America that his extensive notes in the Canadian and Indian notebooks would lead us to believe he was working on…all we have are the notes…more than 4000 pages of them unpublished

I have said that in what Thoreau wrote in these notebooks, as dry and factual as they are, we get a sense of a transition, an empirical turn, that not only occurs in his thought, but that would begin to lay the foundations of modern geography and environmental history…

Besides this however, we may also sense something more important…something deeper that you can all take pride in tonight… Thoreau used many books, articles and maps in these notes, most of which were read in libraries just like the one we celebrate this weekend. In his life and work we can see how something as humble and as common, as book or a map in a library…can spark the imagination and inspire great thoughts…without a library there would be no Henry David Thoreau, there would be no notes and drawings for us to wonder and marvel at this evening… like Thoreau’s notes, libraries, through their collections of books, manuscripts and maps teach us that…

Exploring the Ruins of Tikal...

My article in Alpinist Magazine 41....called Vericality: the other blank on the map....

when not climbing in the Alps, mountain biking through some jungle or looking for Roman ruins in North Africa, is a Specialist in Modern Cartography and Geographic Information Science in the Geography and Map Division of the Library of Congress. A Fellow of the Royal Geographical Society he is the recent recipient of a J. L Heiberg Research and Exploration Fellowship for his work on the physical remains of Roman Centuriation and in 2010 was awarded a J.S. Best Fellowship from the American Geographical Society.
While at the Library of Congress, much of his research has concentrated on the use of computer modeling in the analysis of Roman, Medieval and Renaissance maps.
He is also a lecturer on the history of cartography and and Geographic Information Sciences in the Graduate School for Advanced Studies at the Johns Hopkins University.

Kant in the Wilderness: Thoreau's Geographic Turn

.....click on Henry to read my new article just published by the Thoreau Society

History of GIS...my new article in ArcNews....

...click on above to read.....

Exploring the Mer de Glace...

...and its melting....

In the Land of Melting Glaciers...

...working on medieval land charters and using them to reconstruct early land ownership law....

David Kendall's article "Recovery of Structure From Fragmentary Information" first showed how to use statistical methods to reconstruct geographical layouts and maps from chartularies and medieval deeds....

Kendall's early explorations...

...making maps from fragmentary data...

Kendall working on statistical map reconstuction.....

...I try in my reconstructions to follow in his footsteps....

Recent Appearance on C-Span American History Channel

Click to view video....

Deep in the gorges of Provence....

...all Bruce Chatwin needed was a leather jacket, 20 bucks in his pocket and a notebook and the world was his.....words to live by....

Eric Shipton surveying in the Karakorum....

...if I could only write about it all like he did....

Recent and forthcoming publications....

...another new article...just published in the Portolan....

In Nietzsche's Shadow: Searching for the Remains of Roman Cartography in Southern France

My newest book just released....

..click on the cover to link to the New York Times Review above....

Around the world...some images from recent travels and mapping projects....

Searching for and mapping the megaliths on the island of Bornholm....

" 'Amateur' field geographers can speak with authority about the clarifying effects on the mind of direct physical danger in the real world and there exists a terrible antagonism between field geographers and armchair academics. Not only do those in their armchairs think and write junk, obfuscation, obscurantism, and endlessly convoluted self-referral to their literature in windowless libraries, they do not care about the human condition.”

--William Bunge, Geography is a Field SubjectArea, 1983

It is Bunge’s words that I remember when I am out in the field in places like Tunisia and Algeria.

Exploring theTomb of Cleopatra II, near Tipasa, Algeria

Summer, 2011

...the tomb as it looked in 1926....

...entrance to the tomb....

...the underground aqueducts and sewers of Tipasa, Algeria...

...nothing but questions... trying to map their extent....Summer 2011...

Heidegger Climbing in the Mountains near Davos....

For Peter Gould.... who showed me there was something in Heidegger that geographers could use...

"Indeed space is still one of the things that is constitutive for the world... ---Martin Heidegger

I had read text after text of Martin Heidegger under the tutelage off one of the finest scholars it has ever been my privilege to meet. The philosopher Joseph Kockelmans always worked with parallel German and English texts, and with the Greek text too if it were appropriate, and slowlyopened up the thinking that lay behind Heidegger's tenacious quest. If anyone had told me we would spend sixteen weeks on the forty-five pages of Heidegger's "Anaximander Fragment," I would have laughed, but at the age of forty seven, I began to read properly.---Peter Gould, Becoming a Geographer

...a must read for all geographers and cartographers....

Wittgenstein & Cartographic Theory

...slightly modified [added to] selections from the Tractatus...

We picture facts to ourselves
A picture [map] presents a situation in logical space, the existence and non-existence of states of affairs.
A picture [map] is a model of reality.
A picture [map] is a fact.
A pictorial [cartographic] form is the possibility that things are related to one another in the same way as elements of a picture [cartographic objects].That is how a picture [map] is attached to reality; it reaches right out to it. It is laid against reality like a measure.
Every picture [map] is at the same time a logical[mathematical][spatial] one. Not every picture is however a spatial one. [every map is].It is as impossible to represent in language anything that contradicts logic as it is impossible to represent by its coordinates a figure that contradicts the laws of space [projection], or to give coordinates of a point that does not exist. [void]

...Wittgenstein's grave at Acsension Parish Burial Ground, Cambridge....notice the little ladder that recalls the last few statements of the Tractatus....

...my theory of cartographic history...

The actual maps that exist are but a tiny subset of the theoretical maps that could exist. These real maps are products of a very small number of trajectories through cartographic space...each with its own unique place in this mathematical construction. Every real map is surrounded by a tiny cluster of real or unreal neighbors who are its ancestors and descendents...

...this notion takes its starting point from David Lewis' "On the Plurality of Worlds".

...think of maps as random graphs.....

...both of these books by Tim Robinson are must reads for any cartographer....or historian of maps.

....Robinson's reflections on cartography and the effect of landscape on abstract thought are among the most insightful in all of cartographic literature.....

Gian-Carlo Rota.....

His MIT lectures changed my way of thinking about mathematics....

His "End of Objectivity" lectures were groundbreaking and are currently not easy to come by.....

Galois Lattices....

....maps are represented using attribute tables and the connections between them are calculated using programs like Galacia....click on lattcies for more on Galacia......more details coming in a future full post...

Garrett Birkhoff....

...his Harvard Lectures on Lattice Theory...the class known as MATH 252...developed some of the most beautiful and applicable algebraic structures to the study of stemma and recensions.

The mathematics of the medieval Portolan chart

Click for the Washington Post Story on my modeling of Portolan Charts

Mathematical models of the 1507 Waldseemuller Map

...click to read the Washington Post article on my research....

This blog is featured at the National Library of Scotland's Georeferencing page

Waldseemuller and me....

Click on image for the video "Bridging the Gap Between Physical and Digital Preservation" about the encasing and digitization of the Waldseemuller map...

My Recent Books...

The Naming of America: Martin Waldseemuller's 1507 World Map and the Cosmographiae Introductio

...this is my translation and commentary on Waldseemuller's Cosmographiae Introductio....click on image to order from Amazon...

Reviews of The Naming of America

Imago Mundi:

....Hessler’s nuanced translation brings to life this dynamic period of cartographic history and the theories used by these early sixteenth-century cosmographers.His close attention to the Latin and his extensive notes reflect a level of serious scholarship that should place this book on the reading list for all graduate seminars focused on understanding the production of early modern knowledge. The presentation, graphic aesthetic and accessibility of the text will make this a favourite for general readers, as well, and should be on the wish list of everyone interested in early American history and cartography.

California Literary Review:

...It lurks in the background of our childhood imagination, now and again roaring back in adulthood to remind us of possibilities. A map of the world, that fixture in elementary classrooms, has always been a book masquerading as a flat piece of paper. Like layers of the earth for geologists, maps offer a glinting sample of the past. And when it comes to the Waldseemüller map, the Universalis Cosmographia that forms the subject of The Naming of America by John W. Hessler, there are earth-shattering discoveries to be found. Let it be said, up front, that The Naming of America is not a popular work in the vein of Doris Kearns Goodwin or Stephen Ambrose. Hessler’s is a scholarly affair, impeccably printed, where the footnotes are as long as the text, and controversies are discussed with dry impartiality....

Warntz and Tobler...two mathematical pioneers....

William Warntz on topological and cartographic surfaces....

We now look upon maps not only as stores for spatially ordered information, but also as a means for the graphical solution of certain spatial problems for which the mathematics proves to be intractable, and to produce the necessary spatial transformations for hypothesis testing....The modern geographer concieves of spatial structures and spatial processes as applying not only to such things as landforms....but also to social, economic, and cultural phenomena portraying not only conventional densities but other things such as field quantity potentials, probabilites, refractions etc etc. Always these conceptual patterns may be regarded as overlying the surface of the real earth and the geometrical and topological characteristics of these patterns, as tranformed mathematically or graphically, thus describe aspects of the geography of the real world...

---Spatial Order, Harvard Papers in Theoretical Geography 1, 1967

We recognize yet another role for maps. In the solution of certain problems for which the mathematics, however elegantly stated, is intractable, graphical solutions are possible. This is especially true with regard to "existence theorems". There are many cases in which the graphical solution to a spatial problem turns out to be a map in the full geographical sense of the term, "map." Thus a map is a solution to the problem.

The basic ideas of the theory of convex sets are naturally and easily appreciated when examined in a geographical context, rather than in a non-spatial one...in this respect geography is truly elegant.

Maps showing regional classification can be regarded as logic diagrams. Mapping of sets is a general mathematical concept. Geographical mapping is merely a special case of this.

Tobler's computer program, called Bi-dimensional Regression marks the real beginning of the mathematical analysis of historic maps. The paper was published in 1977 along with a Fortran program which performed an 'empirical transformation regressing an independent plane configuration against a similar configuration.' The program itself constructs this geometric transformation by estimating two sets of bivariate points on a square lattice that have been interpolated from the originally irregularly arranged original observations. This is very much analogous to modern landmark morphometrics using thin-plate splines and various other methods. The curvilinear regression coefficients are represented by Tobler as a spatially varying, but coordinate invariant, second-order tensor field...

Tobler's paper is geometrically insightful and, although being from the 1970's, still contains ideas that are yet to be fully explored in the study of the history of cartography.... Two-dimensional asymmetric tensor analysis for example... his work inspires everything I do....

By far the best theoretical work on the methodology of the history of cartography.

What the historian of cartography should be concerned with is a systematic study of the factors affecting error, and seek to establish their cause and variability and the statistical parameters by which error is characterized.....